"I like this cup."

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I can't answer the different between using "mögen" and "gefallen," but you use dative here because it's literally saying "This cup is pleasing to me" or "This cup appeals to me" - in which "me" is an indirect object i.e. the dative case.

Because of inflection in German, the word order can be switched around without changing the meaning. From what I've gathered, the word order is changed depending on what you would like to emphasize. I don't know enough German yet to really understand how the emphasis works though!